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Discrete Harmonic Analysis Associated with Ultraspherical Expansions

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Abstract

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted ℓp-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator

$$ {\Delta}_{\lambda} f(n):=a_{n}^{\lambda} f(n+1)-2f(n)+a_{n-1}^{\lambda} f(n-1),\quad n\in \mathbb{N}, \lambda >0, $$

where \(a_{n}^{\lambda } :=\{(2\lambda +n)(n+1)/[(n+\lambda )(n+1+\lambda )]\}^{1/2}\), \(n\in \mathbb {N}\), and \(a_{-1}^{\lambda }:=0\). We also prove weighted ℓp-boundedness properties of transplantation operators associated with the system \(\{\varphi _{n}^{\lambda } \}_{n\in \mathbb {N}}\) of ultraspherical functions, a family of eigenfunctions of Δλ. In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.

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Acknowledgements

We woud like to thank the referee for his/her valuable comments which have allowed us to improve the manuscript.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271, La Laguna (Sta. Cruz de Tenerife), Spain

    Jorge J. Betancor, Juan C. Fariña & L. Rodríguez-Mesa

  2. Department of Mathematics, Nazarbayev University, 010000, Astana, Kazakhstan

    Alejandro J. Castro

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  1. Jorge J. Betancor
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Betancor, J.J., Castro, A.J., Fariña, J.C. et al. Discrete Harmonic Analysis Associated with Ultraspherical Expansions. Potential Anal 53, 523–563 (2020). https://doi.org/10.1007/s11118-019-09777-9

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